The historical context is that prior to 230, imperfect moderation would be used as proof that a service could moderate but chose not to. This created an incentive to architecturally design systems in a way that made moderation impossible (or at least strongly encouraged services to pretend moderation was impossible). So Congress enacted Section 230, so partial moderation wouldn't be turned against you.
Kind of funny in this context. But attempts at moderation, even bad ones, are very much in line with the history of the bill. And of course, we're talking about AWS's private right to associate with whom they want, not government action. 230 doesn't really enter into it except as a metaphor of sorts.
Invoking it also doesn't let you reclaim a DoD contract you lost, or restore public opinion of your company.
(also, protip for trying to convince anybody of anything: don't start out with an assumption that the person you're talking to is ignorant. Replying with "read up on..." is always insulting)
Section 230 says that ISPs are not treated as the publisher or speaker of what their users publish or say. That does not mean that prosecutors cannot try to find companies criminally responsible for allowing users to post illegal content on their platforms, especially if they have knowledge of such content. See Backpage. And even if the companies win, the battle itself is quite damaging.
Furthermore, Section 230 says nothing about how the court of public opinion will treat the provider. These companies are thinking about more than just the legal repercussions. If their business suffers because people do not want to do work with companies that support this kind of content, then all the Section 230 protections in the world will not bring in revenue.
You're right with the second paragraph though, in the US de-facto monopolistic companies are allowed to discriminate their customers regardless of public interest. Visa and Mastercard is allowed to ban Wikileaks from receiving any donations at all (thank God for cryptocurrencies) since they posted info about war crimes and Amazon is allowed to ban social networks that doesn't fact check.
That doesn't mean it shouldn't be that way. Public interests needs some protection, even though companies are the suppliers.
A recipe: 1) Create a bunch of accounts on your competitor's platform. 2) Post awful things using those accounts. 3) Screenshot them as proof that said competitor isn't taking moderation seriously. Make sure to do this before the posts are flagged/downvoted. 4) Send screenshots to all of the vendors that supply services to your competitor.
Voila, competition problem solved.
5) Hire a mob and storm the capital of the United States in an act of sedition, linking the planning and encouraging of the attack to your competitor's platform.
I don't blame Twitter, the company for what happened in Portland this summer, which is sort of the entire point of Section 230 protections. I don't like Twitter or Parler or Gab or Facebook for what they've done to civil discourse, but that doesn't mean I think any of them should get their hosting yanked for it.
In fact, it will almost certainly make things worse. All of the attacks on 4chan and 8chan over the years only served to make them more popular - it's like people have amnesia about the Streisand effect.